Setup

Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by $$ f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F}) , $$ for a fixed convex and l.s.c. function $g$ . Here $\mu$ is a (if acceptable $\sigma$-)finite measure on $\mathcal{X}$.

Question

Under what conditions can the interchanging of the $$ \inf_{f \in U}\int_{x \in \mathcal{X}}f^2(x)\mu(dx) = \int_{x \in \mathcal{X}}\inf_{f \in U}f^2(x)\mu(dx). $$ In other words....when can the infimum be taken pointwise?